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Insert « Academic unit» on every page: 1 Go to the menu «Insert» 2 Choose: Date and time 3 Write the name of your faculty or department in the field «Footer» 4 Choose «Apply to all". Parameterized Algorithms Randomized Techniques. Bart M. P. Jansen. August 18th 2014, Będlewo. - PowerPoint PPT Presentation

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Parameterized AlgorithmsRandomized TechniquesBart M. P. JansenInsertAcademic unit on every page:

1 Go to the menu Insert2 Choose: Date and time3 Write the name of your faculty or department in the field Footer4 Choose Apply to all"

August 18th 2014, BdlewoUNIVERSITY OF BERGEN1Randomized computationFor some tasks, finding a randomized algorithm is much easier than finding a deterministic oneWe consider algorithms that have access to a stream of uniformly random bitsSo we do not consider randomly generated inputsThe actions of the algorithm depend on the values of the random bitsDifferent runs of the algorithm may give different outcomes, for the same input2Monte Carlo algorithms3Independent repetitions increase success probability4This lecture5Color coding6

Color codingRandomly assign colors to the input structureIf there is a solution and we are lucky with the coloring, every element of the solution has received a different colorThen find an algorithm to detect such colorful solutionsSolutions of elements with pairwise different colors7The odds of getting lucky8

The Longest Path problem9

Color coding for Longest Path10The dynamic programming table11

A recurrence to fill the table12

Randomized algorithm for Longest Path13Analysis for the Longest Path algorithm14Discussion of color coding15Random separation16The Subgraph Isomorphism problem17

Background18Random 2-coloring of host graphs19

Probability of isolating the pattern subgraph20Randomized algorithm for Subgraph Isomorphism21Chromatic coding2223

How to color24Proper colorings25

Probability of finding a proper coloring26Detecting a properly colored solution (I)27

Detecting a properly colored solution (II)28

12233329derandomization30

Why derandomize?Truly random bits are very hard to come byUsual approach is to track radioactive decayStandard pseudo-random generators might workWhen spending exponential time on an answer, we do not want to get it wrongLuckily, we can replace most applications of randomization by deterministic constructionsWithout significant increases in the running time31How to derandomize32Splitting evenly33

Splitters34Perfect hash families derandomize Longest Path35Universal sets36Coloring families37A randomized algorithm for feedback vertex set38

38The Feedback Vertex Set problem

39Reduction rules for Feedback Vertex Set40How randomization helps4142

Monte Carlo algorithm for Feedback Vertex Set43Monte Carlo algorithm for Feedback Vertex Set44Correctness (I)45Correctness (II)46Discussion47Exercises48Summary49The end